| L(s) = 1 | − 2-s + 1.71·3-s + 4-s − 5-s − 1.71·6-s + 1.86·7-s − 8-s − 0.0551·9-s + 10-s − 2.21·11-s + 1.71·12-s + 13-s − 1.86·14-s − 1.71·15-s + 16-s + 4.52·17-s + 0.0551·18-s − 1.08·19-s − 20-s + 3.20·21-s + 2.21·22-s − 8.23·23-s − 1.71·24-s + 25-s − 26-s − 5.24·27-s + 1.86·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.990·3-s + 0.5·4-s − 0.447·5-s − 0.700·6-s + 0.705·7-s − 0.353·8-s − 0.0183·9-s + 0.316·10-s − 0.667·11-s + 0.495·12-s + 0.277·13-s − 0.498·14-s − 0.443·15-s + 0.250·16-s + 1.09·17-s + 0.0130·18-s − 0.248·19-s − 0.223·20-s + 0.698·21-s + 0.471·22-s − 1.71·23-s − 0.350·24-s + 0.200·25-s − 0.196·26-s − 1.00·27-s + 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 17 | \( 1 - 4.52T + 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 37 | \( 1 + 9.03T + 37T^{2} \) |
| 41 | \( 1 - 0.142T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 + 2.28T + 53T^{2} \) |
| 59 | \( 1 - 3.72T + 59T^{2} \) |
| 61 | \( 1 - 0.177T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6.89T + 83T^{2} \) |
| 89 | \( 1 + 8.72T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.023949326566600478213820558155, −7.84203786107617572321269968318, −6.93656941031596138138118437614, −5.85320201302692365238527580230, −5.16997953410926125925750431449, −3.95098121000873719123485712736, −3.33013847459947770638616071071, −2.34100065934952280296335388819, −1.56706601868242486932187282827, 0,
1.56706601868242486932187282827, 2.34100065934952280296335388819, 3.33013847459947770638616071071, 3.95098121000873719123485712736, 5.16997953410926125925750431449, 5.85320201302692365238527580230, 6.93656941031596138138118437614, 7.84203786107617572321269968318, 8.023949326566600478213820558155
